Diffraction-free beams are by definition localized optical wave packets that remain invariant during propagation. Perhaps the best known example of such a diffraction-free wave is the Bessel beam first predicted theoretically and experimentally demonstrated by Durnin, J. J. Opt Soc. Am. A 4, p. 651 (1987) and Durnin, et al. Phys. Rev. Lett. 58, p. 1499 (1987). Other such non-diffracting wave configurations include, for example, higher-order counterparts, as well as waves based on parabolic cylinder functions as described in J. C. Gutierrez-Vega, et al., Opt. Lett. 25, p. 1793 (2000) and M. A. Bandres, et al. Opt. Lett. 29, p. 44 (2004). In systems that exhibit bidiffraction, normal diffraction in one direction and anomalous in the other, such as photonic crystals and lattices, nondiffracting X-waves and Bessel-like beams are also possible as discussed in J. Lu and J. F. Greenleaf, IEEE Trans. Ultrason. Ferroelectr. Freq. Control 39, p. 19 (1992) and in D. N. Christodoulides et al., Opt. Lett. 29, p. 1446 (2004) and O. Manela et al., Opt. Lett. 30, p. 2611 (2005). Strictly speaking, these solutions convey infinite power, and for this very reason they are “impervious” to diffraction.
If, on the other hand, these diffraction-free beams pass through a finite aperture (are truncated), diffraction eventually takes place. Yet, in such cases, the rate of diffraction can be considerably slowed down depending on the degree of truncation, i.e., how large is the limiting amplitude aperture with respect to the features of the beam. In the case of finite Bessel beams, such effects were first theoretically analyzed by Gori, et al., Opt. Commun. 64, p. 491 (1987).
An important aspect associated with such diffraction/dispersion-free wave packets is their dimensionality. In fact all the above-mentioned solutions exist only in (2+1)D and (3+1)D configurations. The problem becomes more involved in the lowest dimension [e.g., in (1+1)D], which is known to describe the diffraction of planar optical beams or pulse propagation in dispersive optical fibers. Yet, even in this case, dispersion-free Airy wave packets are possible, as first predicted by Berry and Balazs within the context of quantum mechanics. This interesting class of Airy structures is unique in the sense that these beams lack parity symmetry and tend to accelerate during propagation. The acceleration process associated with these beams was later interpreted by Greenberger, Am. J. Phys. 48, p. 256 (1980) on the basis of the equivalence principle. In this latter case with Airy wave packet again associated with an infinite energy. In addition by its very nature, the Airy beam is “weakly confined,” since its oscillating tail decays very slowly. Therefore, for all practical purposed, it is difficult to synthesize such beams unless of course they are amplitude truncated. Finite energy (exponentially decaying) diffractionless Airy planar beams in nonlinear unbiased photorefractive crystals have been predicted as a result of thermal diffusion. Yet, finite power Airy wave packets have never been investigated under linear conditions.
In 1979 M. V. Berry and N. L. Balazs, Am. J. Phys. 47, 264 (1979) made an important observation within the context of quantum mechanics: they theoretically demonstrated that the Schrodinger equation describing a free particle can exhibit a nonspreading Airy wave packet solution. Perhaps the most remarkable feature of this Airy packet is its ability to freely accelerate even in the absence of any external potential. In one dimension, this Airy packet happens to be unique, e.g., it is the only nontrivial solution, apart from a plane wave, that remains invariant with time.
Siviloglou, G, and Christodoulides, D., “Accelerating finite energy Airy beams, Optics Letters, Vol. 32, No. 8, Apr. 15, 2007, pp. 979-981 discusses properties of Airy beams and the investigation of the acceleration dynamics of quasi-diffraction-free finite energy Airy beams and concludes that freely accelerating finite energy Airy beams are possible in both one- and two-dimensional configurations. However, the publication does not teach, and thus is not enabling in regard to methods or systems for implementing Airy wave packets in dispersive optical fibers.
Over the years, nonspreading or nondiffracting wave configurations have been systematically investigated in higher dimensions (two and three dimension), particularly in the areas of optics and atom physics. What makes the analogy between these two seemingly different disciplines possible is the mathematical correspondence between the quantum mechanical Schrodinger equation and the paraxial equation of diffraction. In terms of experimental realization, optics has thus far provided a fertile ground in which the properties of such nonspreading localized waves can be directly observed and studied in detail. Perhaps the best known example of such a two dimensional diffraction-free optical wave is the so-called Bessel beam first suggested and observed and studied in detail. Perhaps the best known example of such a 2D diffraction-free optical wave is the so-called Bessel beam first suggested and observed by Durnin, et al.
Even though at first sight, the aforementioned propagation-invariant beams may appear dissimilar, they in fact share common characteristics. First, they are all generated from an appropriate conical superposition of plane waves. Even more importantly, all these solutions are known to convey infinite power, a direct outcome of their nondiffracting nature. Of course, in practice, all these nonspreading beams are normally truncated by an aperture because of lack of space and power, thus as a result they tend to diffract during propagation. Yet, if the geometrical size of the limiting aperture greatly exceeds the spatial features of the ideal propagation-invariant fields, the diffraction process is considerably “slowed down” over the intended propagation distance and hence for all practical purposes these beams are called “diffraction-free”. No localized one-dimensional propagation-invariant beam can be synthesized through conical superposition.
Quite recently, nonspreading freely accelerating Airy beams have been realized within the context of optics G. A. Siviloglou, et al. Phys. Rev. Lett. 99, p. 213901 (2007). This observation was carried out by exploiting the formal analogy between the free-particle Schrodinger equation and the paraxial equation of diffraction, validating an earlier theoretical prediction by Berry and Balazs, Am. J. Phys. 47, 264, (1979). Perhaps the most intriguing characteristic of the Airy wavepacket is its ability accelerate even in the absence of any external potential discussed in Berry and Balazs and Greenberger, Am. J. Phys. 48, 256 (1980). In fact, in one-dimensional settings the Airy wave happens to be unique. It is important to note that like any other diffraction-free beam, the Airy wave carries infinite power and hence its realization demands some degree of truncation.
In two recent studies, the dynamics of exponentially truncated (finite energy) Airy beams have been explored in Berry and Balazs (1979) and in Siviloglou (2007). In these works it was demonstrated that finite energy Airy beams can still resist diffraction, while their main intensity maxima or lobes tend to accelerate during propagation along parabolic trajectories. The aforementioned acceleration occurs in spite of the fact that center of gravity of these truncated waves remains invariant in accord with Ehrenfest's theorem. This behavior can persist over long distances until diffraction effects eventually take over as described in Siviloglou (2007). Overall these unusual properties of the Airy wavepackets put them in a category by themselves. Unlike other families of nondiffracting fields described in Durnin (1987); Gutierrez-Vega (2000) and Bandres (2004), Airy beams are also possible in one dimension, do not result from conical superposition, and are thus highly assymetric.
In Siviloglou, G, Broky, J., Dogariu, A. and Christodoulides, D., “Observations of Accellerating Airy Beams”, Physical Review Letters, 99, 213901, Nov. 20, 2007, the observations of one- and two-dimensional accelerating diffraction-free Airy beams are disclosed. The observations include that even though Airy beams are exponentially truncated (convey finite power) they still resist diffraction while their main intensity maxima or lobes tend to accelerate during propagation along parabolic trajectories. The publication suggests using a phase mask to generate one- and two-dimensional Airy beams for observing the intensity profile of the Airy beam. The publication concludes that Airy optical wave packets were observed and that they exhibit unusual features such as the ability to remain diffraction free over long distances while they tend to freely accelerate during propagation. Like Siviloglou (Mar. 19, 2007) the publication discloses observations, it fails to provide enabling method or systems for implementing Airy wave packets or applications for using the Airy wave packets.
Thus, what is needed are methods and systems for generating Airy waves and to synthesize such Airy wavepackets in both the spatial and temporal domain.